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Chapter 9: Problem 9

A rod is lying on the top of a table. One end of the rod is hinged to the table so that the rod can rotate freely on the tabletop. Two forces, both parallel to the tabletop, act on the rod at the same place. One force is directed perpendicular to the rod and has a magnitude of 38.0 N. The second force has a magnitude of 55.0 N and is directed at an angle with respect to the rod. If the sum of the torques due to the two forces is zero, what must be the angle ?

Short Answer

Expert verified
The angle \( \theta \) is approximately 43.2 degrees.

Step by step solution

01

Understand Torque

Torque is the measure of the force that can cause an object to rotate around an axis. Torque depends on the force applied, the distance from the pivot (or axis of rotation), and the angle at which the force is applied. It's calculated as \[ \tau = rF \sin \theta \]where \( \tau \) is the torque, \( r \) is the distance from the pivot, \( F \) is the force applied, and \( \theta \) is the angle between the force vector and the lever arm.
02

Set Up Torque Equations

We have two forces acting on the rod leading to two torques: 1. Torque from the first force (38.0 N) is \( \tau_1 = r \cdot 38.0 \sin 90^\circ = r \cdot 38.0\). This is because the force is perpendicular to the rod. 2. Torque from the second force (55.0 N) is \( \tau_2 = r \cdot 55.0 \sin \theta \). Since the sum of the torques is zero, we have:\[ \tau_1 - \tau_2 = 0 \] which implies \[ r \cdot 38.0 - r \cdot 55.0 \sin \theta = 0 \]
03

Simplify the Equation

Since the distance \( r \) is the same and non-zero, it can be canceled from the equation, simplifying to:\[ 38.0 - 55.0 \sin \theta = 0 \]
04

Solve for the Angle

Rearrange the simplified equation to solve for \( \sin \theta \):\[ \sin \theta = \frac{38.0}{55.0} \]Now, calculate \( \theta \) using the inverse sine function:\[ \theta = \sin^{-1} \left( \frac{38.0}{55.0} \right) \]
05

Calculate \( \theta \)

Use a calculator to determine \( \theta = \sin^{-1} \left( \frac{38.0}{55.0} \right) \) which results in \( \theta \approx 43.2^\circ \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rotational Dynamics
Rotational dynamics is the study of how objects rotate and the forces that cause these rotations. When a force is applied to an object at a distance from its center of rotation, it can cause the object to spin or rotate. This spinning motion is influenced by the magnitude of the force, the distance from the pivot point where the force is applied (also called the lever arm), and the angle at which the force is applied. This concept is crucial in understanding the behavior of rotating objects, like wheels or in our case, a hinged rod.
In the context of our exercise, the rod can rotate freely around one fixed end. The dynamics of how the rod rotates depend on how forces are applied to it. By understanding rotational dynamics, we can calculate the angle needed for two forces to keep the rod balanced so that it does not rotate.
Force Vectors
Force vectors are quantities that have both magnitude and direction. In physics, they are used to represent forces acting on an object. The direction of the vector is important because it determines the way the force affects the object's motion or rotation. Forces can be decomposed into components, often using trigonometry, to better analyze the effect each force will have on the system.
In the original problem, we have two forces acting on the rod. One force is perpendicular with a magnitude of 38.0 N, meaning it acts straight across the rod. The other force is 55.0 N, but it acts at an angle relative to the rod. By understanding these vector qualities, we can determine the resultant torque and how these forces interact to maintain equilibrium.
Equilibrium
Equilibrium occurs when all the forces and torques acting on an object are balanced, meaning the object is in a state of rest or moves with constant velocity. In rotational dynamics, this involves the sum of all torques鈥攖he rotational counterparts of forces鈥攂eing zero. If the torques acting on an object cancel each other out, the object won't spin or rotate.
For the rod problem, the condition for equilibrium is that the sum of the torques due to the two forces is zero. By setting the torques to cancel each other鈥攐ne torque positive and the other negative鈥攚e ensure that the rod remains in a stable, non-rotating condition.
Angle Calculation
Angle calculation involves using mathematical tools to determine the precise angles at which forces must act for stability or any desired movement. Often, this involves trigonometric functions like sine, cosine, or tangent, especially when dealing with forces acting at angles.
In the exercise, after setting up the torque equilibrium equation and simplifying, we find that:
  • The sine function is used to resolve the second force at an angle into a component that contributes to torque.
  • We solve for the angle using the inverse sine function: \[\theta = \sin^{-1}\left(\frac{38.0}{55.0}\right)\]
  • Calculating this gives the necessary angle \(\theta\) for equilibrium, which is approximately 43.2 degrees.
Understanding and applying these calculations are key to solving problems involving forces at angles and maintaining equilibrium in rotational dynamics.

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Most popular questions from this chapter

A solid cylindrical disk has a radius of 0.15 m. It is mounted to an axle that is perpendicular to the circular end of the disk at its center. When a 45-N force is applied tangentially to the disk, perpendicular to the radius, the disk acquires an angular acceleration of 120 \(\mathrm{rad} / \mathrm{s}^{2} .\) What is the mass of the disk?

A ceiling fan is turned on and a net torque of 1.8 \(\mathrm{N} \cdot \mathrm{m}\) is applied to the blades. The blades have a total moment of inertia of 0.22 \(\mathrm{kg} \cdot \mathrm{m}^{2}\) . What is the angular acceleration of the blades?

The drawing shows an outstretched arm (0.61 m in length) that is parallel to the floor. The arm is pulling downward against the ring attached to the pulley system, in order to hold the 98-N weight stationary. To pull the arm downward, the latissimus dorsi muscle applies the force \(\overrightarrow{\mathbf{M}}\) in the drawing, at a point that is 0.069 m from the shoulder joint and oriented at an angle of \(29^{\circ} .\) The arm has a weight of 47 \(\mathrm{N}\) and a center of gravity ( cg) that is located 0.28 \(\mathrm{m}\) from the shoulder joint. Find the magnitude of \(\overline{\mathbf{M}} .\)

A platform is rotating at an angular speed of 2.2 rad/s. A block is resting on this platform at a distance of 0.30 m from the axis. The coefficient of static friction between the block and the platform is 0.75. Without any external torque acting on the system, the block is moved toward the axis. Ignore the moment of inertia of the platform and determine the smallest distance from the axis at which the block can be relocated and still remain in place as the platform rotates.

The parallel axis theorem provides a useful way to calculate the moment of inertia \(I\) about an arbitrary axis. The theorem states that \(I=I_{\mathrm{cm}}+M h^{2},\) where \(I_{\mathrm{cm}}\) is the moment of inertia of the object relative to an axis that passes through the center of mass and is parallel to the axis of interest, M is the total mass of the object, and h is the perpendicular distance between the two axes. Use this theorem and information to determine an expression for the moment of inertia of a solid cylinder of radius R relative to an axis that lies on the surface of the cylinder and is perpendicular to the circular ends.
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